Problems

01

Ratio and root tests

Apply the ratio test or the root test to determine whether each of the following series is absolutely convergent, conditionally convergent, or divergent.

(a) ${\displaystyle \sum _{k=0}^{\infty }{\left(\frac{k}{3k+1}\right)}^k}$ $$ (b) ${\displaystyle \sum _{n=1}^{\infty }{(-1)}^n\frac{e^n}{n!}}$ $$ (c) ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{(2n)!}}$

02

Ratio and root tests

Apply the ratio test or the root test to determine whether each of the following series is absolutely convergent, conditionally convergent, or divergent.

(a) ${\displaystyle \sum _{n=1}^{\infty }\frac{{(-2)}^n}{n^{100}}}$ $$ (b) ${\displaystyle \sum _{n=0}^{\infty }{\left(\frac{5n}{10n+4}\right)}^n}$ $$ (c) ${\displaystyle \sum _{n=1}^{\infty }\frac{\sqrt{n}}{3^n}}$